3.2478 \(\int \frac{(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{1}{20} \sqrt{1-2 x} (3 x+2)^2 (5 x+3)^{7/2}-\frac{\sqrt{1-2 x} (18960 x+37439) (5 x+3)^{7/2}}{32000}-\frac{2012291 \sqrt{1-2 x} (5 x+3)^{5/2}}{384000}-\frac{22135201 \sqrt{1-2 x} (5 x+3)^{3/2}}{614400}-\frac{243487211 \sqrt{1-2 x} \sqrt{5 x+3}}{819200}+\frac{2678359321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{819200 \sqrt{10}} \]

[Out]

(-243487211*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/819200 - (22135201*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/614400 - (2012291*S
qrt[1 - 2*x]*(3 + 5*x)^(5/2))/384000 - (Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(7/2))/20 - (Sqrt[1 - 2*x]*(3 + 5*
x)^(7/2)*(37439 + 18960*x))/32000 + (2678359321*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(819200*Sqrt[10])

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Rubi [A]  time = 0.0435996, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {100, 147, 50, 54, 216} \[ -\frac{1}{20} \sqrt{1-2 x} (3 x+2)^2 (5 x+3)^{7/2}-\frac{\sqrt{1-2 x} (18960 x+37439) (5 x+3)^{7/2}}{32000}-\frac{2012291 \sqrt{1-2 x} (5 x+3)^{5/2}}{384000}-\frac{22135201 \sqrt{1-2 x} (5 x+3)^{3/2}}{614400}-\frac{243487211 \sqrt{1-2 x} \sqrt{5 x+3}}{819200}+\frac{2678359321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{819200 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x)^3*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x],x]

[Out]

(-243487211*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/819200 - (22135201*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/614400 - (2012291*S
qrt[1 - 2*x]*(3 + 5*x)^(5/2))/384000 - (Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(7/2))/20 - (Sqrt[1 - 2*x]*(3 + 5*
x)^(7/2)*(37439 + 18960*x))/32000 + (2678359321*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(819200*Sqrt[10])

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx &=-\frac{1}{20} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac{1}{60} \int \frac{\left (-381-\frac{1185 x}{2}\right ) (2+3 x) (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{1}{20} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac{\sqrt{1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac{2012291 \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx}{64000}\\ &=-\frac{2012291 \sqrt{1-2 x} (3+5 x)^{5/2}}{384000}-\frac{1}{20} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac{\sqrt{1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac{22135201 \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx}{153600}\\ &=-\frac{22135201 \sqrt{1-2 x} (3+5 x)^{3/2}}{614400}-\frac{2012291 \sqrt{1-2 x} (3+5 x)^{5/2}}{384000}-\frac{1}{20} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac{\sqrt{1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac{243487211 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{409600}\\ &=-\frac{243487211 \sqrt{1-2 x} \sqrt{3+5 x}}{819200}-\frac{22135201 \sqrt{1-2 x} (3+5 x)^{3/2}}{614400}-\frac{2012291 \sqrt{1-2 x} (3+5 x)^{5/2}}{384000}-\frac{1}{20} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac{\sqrt{1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac{2678359321 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1638400}\\ &=-\frac{243487211 \sqrt{1-2 x} \sqrt{3+5 x}}{819200}-\frac{22135201 \sqrt{1-2 x} (3+5 x)^{3/2}}{614400}-\frac{2012291 \sqrt{1-2 x} (3+5 x)^{5/2}}{384000}-\frac{1}{20} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac{\sqrt{1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac{2678359321 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{819200 \sqrt{5}}\\ &=-\frac{243487211 \sqrt{1-2 x} \sqrt{3+5 x}}{819200}-\frac{22135201 \sqrt{1-2 x} (3+5 x)^{3/2}}{614400}-\frac{2012291 \sqrt{1-2 x} (3+5 x)^{5/2}}{384000}-\frac{1}{20} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac{\sqrt{1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac{2678359321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{819200 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.153989, size = 75, normalized size = 0.5 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (138240000 x^5+615168000 x^4+1229558400 x^3+1505007200 x^2+1362715220 x+1202896557\right )-8035077963 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{24576000} \]

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x)^3*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x],x]

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1202896557 + 1362715220*x + 1505007200*x^2 + 1229558400*x^3 + 615168000*x^4
+ 138240000*x^5) - 8035077963*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/24576000

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Maple [A]  time = 0.008, size = 138, normalized size = 0.9 \begin{align*}{\frac{1}{49152000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -2764800000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-12303360000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-24591168000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-30100144000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -27254304400\,x\sqrt{-10\,{x}^{2}-x+3}-24057931140\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^3*(3+5*x)^(5/2)/(1-2*x)^(1/2),x)

[Out]

1/49152000*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-2764800000*x^5*(-10*x^2-x+3)^(1/2)-12303360000*x^4*(-10*x^2-x+3)^(1/2
)-24591168000*x^3*(-10*x^2-x+3)^(1/2)-30100144000*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1
/11)-27254304400*x*(-10*x^2-x+3)^(1/2)-24057931140*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.51399, size = 147, normalized size = 0.98 \begin{align*} -\frac{225}{4} \, \sqrt{-10 \, x^{2} - x + 3} x^{5} - \frac{4005}{16} \, \sqrt{-10 \, x^{2} - x + 3} x^{4} - \frac{128079}{256} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{1881259}{3072} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{68135761}{122880} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{2678359321}{16384000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{400965519}{819200} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^3*(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

-225/4*sqrt(-10*x^2 - x + 3)*x^5 - 4005/16*sqrt(-10*x^2 - x + 3)*x^4 - 128079/256*sqrt(-10*x^2 - x + 3)*x^3 -
1881259/3072*sqrt(-10*x^2 - x + 3)*x^2 - 68135761/122880*sqrt(-10*x^2 - x + 3)*x - 2678359321/16384000*sqrt(10
)*arcsin(-20/11*x - 1/11) - 400965519/819200*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.75172, size = 331, normalized size = 2.21 \begin{align*} -\frac{1}{2457600} \,{\left (138240000 \, x^{5} + 615168000 \, x^{4} + 1229558400 \, x^{3} + 1505007200 \, x^{2} + 1362715220 \, x + 1202896557\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{2678359321}{16384000} \, \sqrt{10} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^3*(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/2457600*(138240000*x^5 + 615168000*x^4 + 1229558400*x^3 + 1505007200*x^2 + 1362715220*x + 1202896557)*sqrt(
5*x + 3)*sqrt(-2*x + 1) - 2678359321/16384000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x
+ 1)/(10*x^2 + x - 3))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**3*(3+5*x)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 2.11436, size = 109, normalized size = 0.73 \begin{align*} -\frac{1}{122880000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (108 \,{\left (16 \,{\left (20 \, x + 41\right )}{\left (5 \, x + 3\right )} + 2903\right )}{\left (5 \, x + 3\right )} + 2012291\right )}{\left (5 \, x + 3\right )} + 110676005\right )}{\left (5 \, x + 3\right )} + 3652308165\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 40175389815 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^3*(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-1/122880000*sqrt(5)*(2*(4*(8*(108*(16*(20*x + 41)*(5*x + 3) + 2903)*(5*x + 3) + 2012291)*(5*x + 3) + 11067600
5)*(5*x + 3) + 3652308165)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 40175389815*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x +
3)))